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We investigate the two-component BKP hierarchy equation for its Lie point symmetries. To obtain a complete classification of the group-invariant solution, we derive the one-dimensional optimal system of subalgebras of A 3 , 3 (D ⨂ s T 2). By employing those subalgebras, we construct the invariant solutions which are represented through Weierstrass elliptic functions, Jacobi elliptic functions, and soliton solutions. Also, we analyse the existence of a bounded travelling-wave solution for the equation. Moreover, through the utilization of scale-invariant symmetry, we derive a self-similar solution. Additionally, by conducting singularity analysis, the solution can be expressed in the form of a right Painlevè series. • The two-component BKP hierarchy equation is studied under the lens of Lie symmetry. • We find the Lie point symmetries and perform a symmetry reduction of this equation. • Group invariant solutions are constructed. [ABSTRACT FROM AUTHOR]

Let f \colon X \to B be a nonisotrivial complex elliptic surface and let \mathcal {D} \subset X be an integral divisor dominating B. We study finiteness related properties of generalized (S, \mathcal {D})-integral sections \sigma \colon B \to X of X where S \subset B is arbitrary. These integral sections \sigma are algebraic and satisfy the geometric condition f (\sigma (B) \cap \mathcal {D})\subset S. For S \subset B finite, the set of (S, \mathcal {D})-integral sections of X is finite by the well-known Siegel theorem. In this article, we establish a general quantitative finiteness result for large unions of (S, \mathcal {D})-integral sections in which both the subset S and the divisor \mathcal {D} are allowed to vary in families where notably S is not necessarily finite or countable. Some applications to generalized unit equations over function fields are also obtained. [ABSTRACT FROM AUTHOR]

The general solutions for nonlinear waves in a bubble-liquid mixture are obtained from the Kudryashov–Sinelshchikov (KS) equation under different parametric regimes. The Chiellini integrability condition has been used for constructing exact general solutions. In the non-dissipative case, we find that the solutions may be expressed in terms of Jacobi elliptic and Weierstrass functions. On the other hand, in the dissipative case, only implicit solutions are obtained for the KS equation. For an alternative perspective, the notion of the Jacobi Last Multiplier has been utilized to obtain the corresponding Lagrangian and Hamiltonian description of the reduced equation for the bubble-liquid mixture system. [ABSTRACT FROM AUTHOR]

Models attempting an analytical description of free-electron laser (FEL) devices have been proposed in the past. They provided interesting results, leading either to a deeper understanding of the FEL dynamics and to semi-analytical formulae, useful for the preliminary design of self amplified spontaneous emission and oscillator FELs. Most of these models work well until the level of mild saturation. In this paper, we comment on the so-called logistic model and a more recent analysis describing the FEL evolution in terms of Jacobi elliptic functions. Both models are shown to be suited to describe the evolution from the low signal to the onset of saturation. We attempt therefore an extension of these theoretical formulations using a delayed logistic model, capable of including characteristic features like the post saturation power oscillations. [ABSTRACT FROM AUTHOR]

This paper thoroughly investigates the integrable Akbota equation, a Heisenberg ferromagnet-type equation that plays a crucial role in exploring curve and surface geometry. The ϕ6-model expansion method, known for its proficiency and reliability, is applied to generate generalized solitonic wave profiles, spanning a diverse range of soliton families. Prior to this study, there is no existing study in which this technique is utilized that ensured the existence of solution via development of conditions. On the other hand, this method is provided with the Jacobi elliptic function-based solutions with the limiting cases. The analytical strategy presented a notable advantage by specifying a constraint for each solution, ensuring its existence. Consequently, the solitonic wave structures exhibit various attributes, including the Jacobi elliptic function, periodicity, brightness, dark-brightness, singularity, exponential, trigonometry, and rational solitonic structures. These characteristics emerge under previously unexplored existence conditions. The results are visually depicted through 2D, 3D, and contour plots, providing a clear illustration of the behavioral responses to pulse propagation and allowing for the inference of fitting values for system parameters. This visualization offers valuable insights into the characteristics and dynamics of soliton solutions derived from the integrable Akbota equation. [ABSTRACT FROM AUTHOR]

In this study, we apply the extended direct method to seek new similarity solutions and reduction equations of the (2+1)-dimensional Kadomtsev–Petviashvili (KP) equation. Through this method, we can reduce the KP equation to a system of ordinary differential equations, and we divide the discussion into four cases via calculation. In some cases, new similarity reductions and exact solutions are obtained, such as elliptic function solutions, polynomial solutions. In other cases, similarity solutions are consistent with the present solutions. Futhermore, a similarity solution we get has more clear and concise form than that obtained by the generalized Clarkson–Kruskal direct method. Therefore, it demonstrates that our study is correct and more efficient than before which can also be applied to the other nonlinear physical models. [ABSTRACT FROM AUTHOR]

We investigated the extended sixth-order Korteweg–de Vries (KdV6) equation which describes many nonlinear phenomena. The improved modified (IM) extended tanh-function method is used to conduct the study. Many exact solutions, like dark, combo singular-dark and singular soliton are achieved. Moreover, hyperbolic solutions, singular periodic solutions, rational solutions, exponential solutions and Jacobi elliptic function (JEF) solutions are derived. Graphical illustrations are presented to visually depict the dynamics of selected solutions. [ABSTRACT FROM AUTHOR]

Despite the historical position of the F-expansion method as a method for acquiring exact solutions to nonlinear partial differential equations (PDEs), this study highlights its superiority over alternative auxiliary equation methods. The efficacy of this method is demonstrated through its application to solve the convective–diffusive Cahn–Hilliard (cdCH) equation, describing the dynamic of the separation phase for ternary iron alloys (Fe–Cr–Mo) and (Fe–X–Cu). Significantly, this research introduces an extensive collection of exact solutions by the auxiliary equation, comprising fifty-two distinct types. Six of these are associated with Weierstrass-elliptic function solutions, while the remaining solutions are expressed in Jacobi-elliptic functions. I think it is important to emphasize that, exercising caution regarding the statement of the term 'new,' the solutions presented in this context are not entirely unprecedented. The paper examines numerous examples to substantiate this perspective. Furthermore, the study broadens its scope to include soliton-like and trigonometric-function solutions as special cases. This underscores that the antecedently obtained outcomes through the recently specific cases encompassed within the more comprehensive scope of the present findings. [ABSTRACT FROM AUTHOR]

The Landau–Ginzburg–Higgs equation is one of the significant evolution equation in physical phenomena. In this work, the exact solutions of this equation are gained by applying an analytical method depends on twelve Jacobi elliptic functions. This equation is turned into an ordinary differential equation by the proposed method. When solving the Landau–Ginzburg–Higgs equation, an auxiliary ordinary differential equation is considered. Some theorems and corollaries utilized in the solutions of this auxiliary equation are given. Using these solutions, the elliptic and elementary solutions of the Landau–Ginzburg–Higgs equation are obtained and illustrated by tables. Many solutions are given in the form of the complex, rational, hyperbolic, and trigonometric functions. The soliton solutions and the complex valued solutions are also found by proposed method. These solutions include the largest set of solutions in the literature. Some of them are shown graphically by 2-dimensional and 3-dimensional with the help of Mathematica software. The obtained solutions are beneficial for the farther development of a concerned model. The presented method does not need initial and boundary conditions, perturbation, or linearization. Besides, this method is easy, efficient, and reliable for solutions of many partial differential equations. [ABSTRACT FROM AUTHOR]

It is shown that if the equation f (z + 1) n = R (z , f) , where R(z, f) is rational in both arguments and deg f (R (z , f)) ≠ n , has a transcendental meromorphic solution, then the equation above reduces into one out of several types of difference equations where the rational term R(z, f) takes particular forms. Solutions of these equations are presented in terms of Weierstrass or Jacobian elliptic functions, exponential type functions or functions which are solutions to a certain autonomous first-order difference equation having meromorphic solutions with preassigned asymptotic behavior. These results complement our previous work on the case deg f (R (z , f)) = n of the equation above and thus provide a complete difference analogue of Steinmetz' generalization of Malmquist's theorem. [ABSTRACT FROM AUTHOR]

The goal of this paper is to find exact solutions to the improved modified Korteweg-de Vries (mKdV) equation with a conformable derivative using the Jacobi elliptic function expansion method. The improved mKdV equation is a prominent mathematical model in the realm of nonlinear partial differential equations, with widespread applicability in diverse scientific and engineering domains. This study leverages the well-known effectiveness of the Jacobi elliptic function expansion method in solving nonlinear differential equations, specifically focusing on the intricacies of the improved mKdV problem. The investigation reveals innovative and explicit solutions, providing insight into the dynamics of the related physical processes. This paper provides a comprehensive examination of these solutions, emphasizing their distinct features and depictions using Jacobi elliptic functions. These findings are especially advantageous for specialists in the fields of nonlinear science and mathematical physics, providing significant insights into the behavior and development of nonlinear waves in various physical situations. This work also contributes to our knowledge of the improved mKdV equation and shows that the Jacobi elliptic function expansion method is a useful tool for solving complex nonlinear situations. The study is enhanced with graphical illustrations of various solutions, which further enhance its analytical complexity. [ABSTRACT FROM AUTHOR]

This study addresses the integrable Akbota equation which is a Heisenberg ferromagnet-type equation and has a significant study of the curve and surface geometry. The conservation laws of the model are computed using the multipliers approach. The Jacobi elliptic function solutions for the model are derived by utilizing the new extended auxiliary equation method. The Jacobi elliptic functions solutions are degenerated to dark, bright, singular and singular periodic solitons in their limit conditions. Further, the modulation instability analysis of the equation is studied. Physical interpretations of the obtained results are represented graphically. To the best of our knowledge, the obtained results in this article are novel and have not been reported before. [ABSTRACT FROM AUTHOR]

In this paper, the improved Fan subequation method is employed to construct exact traveling wave solutions for a generalized fifth‐order dispersive equation. By making use of bifurcation theory of dynamical systems, we have succeeded to obtain the phase portraits of the subequations involving all possible parameter conditions, and many different types of traveling wave solutions as well, which include more general soliton solutions, kink solutions, triangular function solutions, rational solutions, and Jacobian elliptic function solutions with double periods and so on. Furthermore, as all parameters in the representations of exact solutions are free variables, the solutions obtained show more complex dynamical behaviors and could be applicable to explain diversity in qualitative features of wave phenomena. [ABSTRACT FROM AUTHOR]

In this article, we study for the first time the stochastic Kaup–Newell equation in birefringent fibers. Three integration algorithms are applied, namely, the unified Riccati equation expansion method, the addendum to Kudryashov's method, and the addendum to modified sub‐ODE method. Many types of optical soliton solutions such as Jacobi‐elliptic function solutions, Weierstrass elliptic function solutions, bright solitons, dark solitons, singular solitons, and periodic function solutions are obtained. Numerical schemes give a visual perspective to the solutions derived analytically. [ABSTRACT FROM AUTHOR]

In this work, newly created soliton solutions for ion sound and Langmuir waves with an Atangana–Baleanu fractional (ABF) are given. Symbolic software is used to perform the unified solver method (USM) and Weierstrass elliptic function method (WEFM) in order to solve this model. In terms of hyperbolic functions, extended trigonometric functions, and so forth. Single-wave solutions that were entirely novel and universal were attained. The way the soliton solutions behaved in connection with the two-dimensional (2D) and three-dimensional (3D) graphics was also investigated. The dynamic investigation of the newly created soliton solutions reveals that they have several soliton forms like single-soliton, bell-shaped and mixed-form soliton profiles. This strategy is viewed as promising for handling a range of ABF evolution systems. [ABSTRACT FROM AUTHOR]

The present work focuses on the nonlinear dynamics of the synchronous oscillating multiple bubbles in two typical spatial locations, namely, cuboid and rectangle arrangements. The governing equation for such synchronous oscillating multiple bubbles is derived from a modified Rayleigh–Plesset equation. Theoretical results including the collapse time and analytical solution (in three forms) for multiple vapor bubbles, as well as the maximum/minimum radii, oscillation period, and analytical solution in the form of Weierstrass elliptic function for multiple gas-filled ones, are provided. On the basis of these results, we not only study the dynamic characteristics of multi-bubbles straightforwardly but also carefully observe a series of evolution behaviors of bubbles when the number of bubbles decreases gradually on the order of 8 → 4 → 2 → 1. It should be pointed out that we also compare the multi-bubble behaviors between the general cuboid/rectangle arrangements and the corresponding cube/square arrangements under two reasonable restrictions, respectively. Furthermore, the limiting behaviors of the synchronous oscillating multiple gas-filled bubbles are discussed as the initial pressure of the gas in bubble approaches to zero. [ABSTRACT FROM AUTHOR]

The extended Kawahara (Gardner Kawahara) equation is the improved form of the Korteweg–de Vries (KdV) equation, which is one of the most significant nonlinear evolution equations in mathematical physics. In that research, the analytical solutions of the conformable fractional extended Kawahara equation were acquired by utilizing the Jacobi elliptic function expansion method. The given expansion method was applied to different fractional forms of the extended Kawahara equation, such as the fraction that occurs in time, space, or both time and space by suitably changing the variables. In addition, various types of fractional problems are exhibited to expose the realistic application of the given method, and some of the obtained solutions were illustrated in two- or three-dimensional graphics as proof of the visualization. [ABSTRACT FROM AUTHOR]

For the purpose of constructing the exact periodic solutions of nonlinear wave equations, it has been proposed to use a method known as the Jacobi elliptic function expansion method. This method is more general than the hyperbolic tangent function expansion method. It has been demonstrated that the periodic solutions obtained using this method contain both solitary wave solutions and shock wave solutions in some instances. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the nonlinear longitudinal wave equation (LWE) which involves mathematical physics in a magneto-electro-elastic (MEE) circular rod. By three different methods, we found that various forms for exact traveling wave solutions of longitudinal bud equation among an MEE round rod. These new exact and solitary wave solutions are derived in the form of rational, single periodic function, double periodic Jacobian elliptic functions (JEF), especially about Weierstrass elliptic function (WEF). Additionally, certain interesting (3D), (2D) figures represent the LWE provides the corporal information to explain the physically phenomena. By choosing different suitable values of the free parameters, we made some charts and we analyzed these charts to get a lot of valuable information to understand the MEE circular rod. Our work describes the dynamics of the MEE circular rod. Finally, we conclude with some perspectives for our future research work. [ABSTRACT FROM AUTHOR]

The exact traveling wave solutions to coupled KdV equations with variable coefficients are obtained via the use of quadratic Jacobi's elliptic function expansion. The presented coupled KdV equations have a more general form than those studied in the literature. Nine couples of quadratic Jacobi's elliptic function solutions are found. Each couple of traveling wave solutions is symmetric in mathematical form. In the limit cases m → 1 , these periodic solutions degenerate as the corresponding soliton solutions. After the simple parameter substitution, the trigonometric function solutions are also obtained. [ABSTRACT FROM AUTHOR]

The coupled-Higgs equation, which depicts a system of preserved scalar nucleons cooperating with unbiased scalar mesons, is investigated using the ϕ 6 -model expansion approach in this paper. We want to look at the various particles that are created when scalar nucleons collide with unbiased scalar mesons. The modulus of the elliptic function m → 0 and m → 1 is used to obtain the trigonometric and hyperbolic functions. Under the stated limited constraints, all found solutions are used to validate the partial differential equation. We displayed 3D surfaces and 2D visuals of selected solutions produced with the assistance of a computer program to shed more light on the dynamic behavior of the acquired solutions. [ABSTRACT FROM AUTHOR]

This research examines the fractional Peyrard–Bishop DNA dynamical governing system, which displays the proliferation of optical pulses in field of plasma and the optical fibre. The analytical method is utilized to find travelling wave solutions because the inverse scattering transform cannot solve the Cauchy problem for this equation. The Φ 6 -model expansion method is an efficient and dependable technique for generating the generalised solitonic wave profiles with a wide range of soliton families. The main advantage of the offered analytical strategy is that it specifies a constraint for each solution to guarantee its existence. As a result, solitonic wave structures get attributes such as the Jacobi elliptic function, periodicity, brightness, dark-brightness, singularity, exponential, trigonometry, and rational solitonic structures, among others, under existence conditions that have not been explored previously. The results are represented graphically in 2-D, 3-D, and contour glimpses to illustrate the behavioural responses to pulse propagation by inferring the fitting values of system parameters. The stabilty of the considered model is discussed and develope the stability condition. The fractional parameter is responsible for reducing singularity and continuing to increase the smoothness in wave patterns. It is easy to employ the Φ 6 -model expansion method to other complicated non-linear systems and acquire solitary waves pattern. [ABSTRACT FROM AUTHOR]

In this note we give an elementary proof of the space-like real-analyticity of solutions to a degenerate evolution problem that arises in the study of fractional parabolic operators of the type (\partial _t - \operatorname {div}_x(B(x)\nabla _x))^s, 0

In this paper, we use a new, extended Jacobian elliptic function expansion method to explore the exact solutions of the (2+1)-dimensional asymmetric Nizhnik–Novikov–Veselov (aNNV) equation, which is a nonlinear physical model to describe an incompressible fluid. Combined with the mapping method, many new types of exact Jacobian elliptic function solutions are obtained. As we use two new forms of transformation, most of the solutions obtained are not found in previous studies. To show the complex nonlinear wave phenomena, we also provide various wave structures of a group of solutions, including periodic wave and solitary wave structures of ordinary traveling wave solutions, horseshoe-type wave, s-type wave and breaker-wave structures superposed by two kinds of waves: chaotic wave structures with chaotic behavior and spiral wave structures. The results show that this method is effective and powerful and can be used to construct various exact solutions for a wide range of nonlinear models and complex nonlinear wave phenomena in mathematical and physical research. [ABSTRACT FROM AUTHOR]

Using the traveling wave transformation, the seventh-order KdV equation reduces to a sixth-order complex differential equation (CDE), and we first prove that all meromorphic solutions of the CDE belong to the class W via Nevanlinna's value distribution theory. Then abundant new meromorphic solutions of the sixth-order CDE have been established in the finite complex plane with the aid of an extended complex method and Painlevé analysis, which contains Weierstrass elliptic function solutions and exponential function solutions, some of them are whole new solutions comparing to the opening literature. We give the computer simulations of some elliptic and exponential solutions. At last, we investigate the meromorphic solutions of the nonlinear dispersive Kawahara equation as an application. [ABSTRACT FROM AUTHOR]

In this paper, we consider the (3 + 1)-dimensional fractional-stochastic quantum Zakharov–Kuznetsov equation (FSQZKE) with M-truncated derivative. To find novel trigonometric, hyperbolic, elliptic, and rational fractional solutions, two techniques are used: the Jacobi elliptic function approach and the modified F-expansion method. We also expand on a few earlier findings. The extended quantum Zakharov–Kuznetsov has practical applications in dealing with quantum electronpositron–ion magnetoplasmas, warm ions, and hot isothermal electrons in the presence of uniform magnetic fields, which makes the solutions obtained useful in analyzing a number of intriguing physical phenomena. We plot our data in MATLAB and display various 3D and 2D graphical representations to explain how the stochastic term and fractional derivative influence the exact solutions of the FSEQZKE. [ABSTRACT FROM AUTHOR]

In the present study, the optimality approach is applied to find the exact solution of the Landau-Ginzburg-Higgs Equation (LGHE) using new transformations. This method is a direct algebraic method for obtaining exact solutions of nonlinear differential equations. We find suitable solutions of the LGHE in terms of elliptic Jacobi functions by applying transformations of basic functions. Exact solutions of the equations are obtained with the help of symbolic software (Maple) which allows the computation of equations with parameter constants. It is exposed that PIM is influential, suitable, and shortest and offers an exact solution of LGHE. [ABSTRACT FROM AUTHOR]

In this paper, we consider the (4+1)-dimensional fractional Fokas equation (FFE) with an M-truncated derivative. The extended tanh–coth method and the Jacobi elliptic function method are utilized to attain new hyperbolic, trigonometric, elliptic, and rational fractional solutions. In addition, we generalize some previous results. The acquired solutions are beneficial in analyzing definite intriguing physical phenomena because the FFE equation is crucial for explaining various phenomena in optics, fluid mechanics and ocean engineering. To demonstrate how the M-truncated derivative affects the analytical solutions of the FFE, we simulate our figures in MATLAB and show several 2D and 3D graphs. [ABSTRACT FROM AUTHOR]

In this paper, we study the spectrum σ(L) of the complex Hill operator with Darboux-Treibich-Verdier potential L =d²/dx² - 6P(x + z0; σ) - 2P(x +1/2 + z0; σ) in L²(R,C), where P(z; σ) is the Weierstraß elliptic function with periods 1 and σ, and z0 㼰 C is chosen such that L has no singularities on R. We give a complete picture of the deformation of the spectrum with σ = 1/2 + ib as b > 0 varies. A new idea of the proof is to apply the result of the mean field equation and its connection with this operator. [ABSTRACT FROM AUTHOR]

The Fokas-Lenells (FL) equation, which is rich in physical property in soliton theory as well as optical fiber, is a generalization of the higher-order Schrödinger equation. We construct the periodic solutions of the FL equation based on the Jacobi elliptic function expansion method in this context. Moreover, the characteristics of the obtained solutions are visualized graphically by selecting appropriate parameters. [ABSTRACT FROM AUTHOR]

This paper deals with a unified presentation traveling wave solutions for the Mukherjee–Kundu equation which is a variant of the well-known nonlinear Schrödinger equation. A Weierstrass function approach has been employed to derive a Weierstrass elliptic function solutions, which can be reduced to the Jacobian elliptic function solutions through the conversion transformations. Due to the reductions of the Jacobian elliptic function solutions, the solitary wave solution and the trigometric period wave solution can be found by m → 1 and m → 0 , respectively. In addition, the features of the obtained traveling wave solutions are represented graphically with the appropriate parameters. [ABSTRACT FROM AUTHOR]

In our paper we apply the Jacobi elliptic function expansion method to obtain solutions to the Kundu–Mukherjee–Naskar equation, which is asymmetric in the two transverse directions. We obtain that the solutions contain a quadratic dependency in the phase, i.e. chirp, in one of the two directions. Unlike in previous applications of the method, the chirp does not affect the amplitude of the solutions. [ABSTRACT FROM AUTHOR]

The Painlevé-IV equation has two families of rational solutions generated, respectively, by the generalized Hermite polynomials and the generalized Okamoto polynomials. We apply the isomonodromy method to represent all of these rational solutions by means of two related Riemann–Hilbert problems, each of which involves two integer-valued parameters related to the two parameters in the Painlevé-IV equation. We then use the steepest-descent method to analyze the rational solutions in the limit that at least one of the parameters is large. Our analysis provides rigorous justification for formal asymptotic arguments that suggest that in general solutions of Painlevé-IV with large parameters behave either as an algebraic function or an elliptic function. Moreover, the results show that the elliptic approximation holds on the union of a curvilinear rectangle and, in the case of the generalized Okamoto rational solutions, four curvilinear triangles each of which shares an edge with the rectangle; the algebraic approximation is valid in the complementary unbounded domain. We compare the theoretical predictions for the locations of the poles and zeros with numerical plots of the actual poles and zeros obtained from the generating polynomials, and find excellent agreement. [ABSTRACT FROM AUTHOR]

In this paper, the optical solitons for the complex Ginzburg–Landau equation with Kudryashov's law of refractive index are established. An improved modified extended tanh–function technique is used to extract numerous solutions. Bright and dark solitons, as well as singular soliton solutions, are achieved. In addition, as the modulus of ellipticity approaches unity or zero, solutions are formulated in terms of Jacobi's elliptic functions, which provide solitons and periodic wave solutions. [ABSTRACT FROM AUTHOR]

In this paper we derive approximate analytical solution to a damped and forced oscillator with power law nonlinear restoring force. We illustrate the obtained results with concrete examples. [ABSTRACT FROM AUTHOR]

In the present paper, the improved modified extended tanh-function method is employed to get new exact traveling wave solutions for the doubly dispersive model. Several types of solutions are obtained using the proposed method. Theses solutions are bright, dark and bright-dark combo soliton solutions. Also, hyperbolic type solutions, singular periodic wave solutions, Jacobi elliptic function solutions, exponential solutions and Weierstrass elliptic doubly periodic solutions are obtained. Some of these solutions are reported in this paper for first time, namely, dark solitons, Jacobi elliptic function solutions and Weierstrass elliptic doubly periodic solutions. Furthermore, 3D graphs of some solutions are introduced for illustrating the physical interpretation. [ABSTRACT FROM AUTHOR]

This article considers the stochastic fractional Radhakrishnan-Kundu-Lakshmanan equation (SFRKLE), which is a higher order nonlinear Schrödinger equation with cubic nonlinear terms in Kerr law. To find novel elliptic, trigonometric, rational, and stochastic fractional solutions, the Jacobi elliptic function technique is applied. Due to the Radhakrishnan-Kundu-Lakshmanan equation's importance in modeling the propagation of solitons along an optical fiber, the derived solutions are vital for characterizing a number of key physical processes. Additionally, to show the impact of multiplicative noise on these solutions, we employ MATLAB tools to present some of the collected solutions in 2D and 3D graphs. Finally, we demonstrate that multiplicative noise stabilizes the analytical solutions of SFRKLE at zero. [ABSTRACT FROM AUTHOR]

In this article, we prove that the 〈 p , q 〉 condition holds, first by using the Fuchs index of the complex Kawahara equation, and then proving that all meromorphic solutions of complex Kawahara equations belong to the class W. Moreover, the complex method is employed to get all meromorphic solutions of complex Kawahara equation and all traveling wave exact solutions of Kawahara equation. Our results reveal that all rational solutions u r (x + ν t) and simply periodic solutions u s , 1 (x + ν t) of Kawahara equation are solitary wave solutions, while simply periodic solutions u s , 2 (x + ν t) are not real-valued. Finally, computer simulations are given to demonstrate the main results of this paper. At the same time, we believe that this method is a very effective and powerful method of looking for exact solutions to the mathematical physics equations, and the search process is simpler than other methods. [ABSTRACT FROM AUTHOR]

The Gardner's equation is the generalization of Kortweg-de Vries equation and modified Kortweg-de Vries equation having real life applications in different fields of science, such as, hydrodynamics, quantum field theory, etc. Moreover, it has applications in fluid mechanics where it explains surface and internal waves. This manuscript addresses application of the new ϕ 6 -model expansion method to Gardner's equation for getting its solitary wave solutions. Such new results aid in getting noteworthy advances in various disciplines of science and technology. Periodic, kink, anti-kink, bright, dark, w-shaped and singular soliton solutions of the governing equation have been constructed. The proposed method takes into account a variety of closed form solutions using the Jacobi elliptic functions. The graphical illustration of some of the obtained solutions has been presented for a suitable choice of free parameters to depict the dynamic nature of the obtained wave solutions. [ABSTRACT FROM AUTHOR]

The stochastic fractional (2 + 1)-dimensional Heisenberg ferromagnetic spin chain equation (SFHFSCE), which is driven in the Stratonovich sense by a multiplicative Wiener process, is considered here. The analytical solutions of the SFHFSCE are attained by utilizing the Jacobi elliptic function method. Various kinds of analytical fractional stochastic solutions, for instance, the elliptic functions, are obtained. Physicists can utilize these solutions to understand a variety of important physical phenomena because magnetic solitons have been categorized as one of the interesting groups of non-linear excitations representing spin dynamics in semi-classical continuum Heisenberg systems. To study the impact of the Wiener process on these solutions, the 3D and 2D surfaces of some achieved exact fractional stochastic solutions are plotted. [ABSTRACT FROM AUTHOR]

In this study, some new hypotheses and techniques are presented to obtain some new analytical solutions (localized and periodic solutions) to the generalized Kawahara equation (gKE). As a particular case, some traveling wave solutions to both Kawahara equation (KE) and modified Kawahara equation (mKE) are derived in detail. Periodic and soliton solutions to this family are obtained. The periodic solutions are expressed in terms of Weierstrass elliptic functions (WSEFs) and Jacobian elliptic functions (JEFs). For KE, some direct and indirect approaches are carried out to derive the periodic and localized solutions. For mKE, two different hypotheses in the form of WSEFs are used to derive the periodic and localized solutions. Also, the cnoidal wave solutions in the form of JEFs are obtained. As a realistic physical application, the solutions obtained can be dedicated to studying many nonlinear waves that propagate in plasma. [ABSTRACT FROM AUTHOR]

In this article, with the aid of Maple software, the exact solutions to the space-time fractional symmetric regularized long wave (SRLW) equation are successfully examined by G ′ / G 2 -expansion and extended complex methods. Consequently, three types of traveling wave solutions are found such as Weierstrass double periodic elliptic functions, simply periodic functions, and the rational function solutions. The obtained results will play an important role in understanding and studying SRLW equation. It is easy to see that the extended complex and G ′ / G 2 -expansion methods are reliable and will be used extensively to seek exact solutions of any other fractional nonlinear partial differential equations (FNPDE). [ABSTRACT FROM AUTHOR]

In this paper, the analytical solution that describes the evolution in time of the fractal damped Duffing equation subjected to external forces of elliptic type is derived using He's two-scale fractal transform and the elliptic balance method (EBM). This solution predicts the evolution in time of the Duffing equation and unveils qualitative and quantitative system behavior when the values of the fractal parameter varies, and how these affect the frequency, the wavelength, and the oscillation amplitude from the start of the motion. Comparison of the amplitude–time response curves over the selected time-interval with those obtained from numerical simulations confirms the accuracy of the derived analytical solution. [ABSTRACT FROM AUTHOR]

In this paper, a special class of exact solutions of the coupled higher-dimensional Klein–Gordon–Zakharov (KGZ) equation with power law nonlinearities is constructed explicitly by separation transformation method. One merit of this work is that when the dimension is larger than one, there is an arbitrary function f (ς) , which is used to search for more general exact solutions of the coupled higher-dimensional KGZ equation. As a result, the Jacobi elliptic function solutions and soliton solutions with free parameters are derived, which lead to rich localized nonlinear structures of the coupled higher-dimensional KGZ equation with power law nonlinearities. [ABSTRACT FROM AUTHOR]

In this study, we proved that damped quadratic nonlinear oscillators similar to Duffing and Helmholtz–Duffing damped equations which emerge in bubble dynamics with time-periodic straining flows and solitary-like wave's dynamics may be derived from a new functional approach based on nonstandard Lagrangians and fractional frictions. The solutions of these equations are given in terms of the Jacobi elliptic functions. It was observed that the dynamical model constructed in this study is comparable to dynamical systems with natural Lagrangians for which the Riemann structure is conformally flat which has important implications in dynamical systems with position-dependent mass. [ABSTRACT FROM AUTHOR]

In this article, we construct multi-wave, homoclinic breather, M-shaped rational and periodic cross kink wave solutions for coupled-Higgs equation (CHE) by using distinct transformations. We obtain these solutions with the aid of logarithmic transformations and symbolic computations. Moreover, we also derive some soliton wave solutions for CHE in polynomial forms. These solutions include solitary wave, soliton wave and Jacobi elliptic function solutions and are found by implementing unified technique. We will also discuss the graphical structure of our newly achieved results. [ABSTRACT FROM AUTHOR]

In this work, an approximate analytic solution which is called a semi-analytical solution to the pendulum equation is obtained. Moreover, the semi-analytical solution is compared to the approximate numerical solution obtained with the aid of Runge-Kutta method. [ABSTRACT FROM AUTHOR]

In this paper, we give the analytical solution and an approximate trigonometric solution to the undamped and unforced Duffing-Helmholtz equation. We also provide high accurate formulas to approximate the Weierstrass elliptic function and its period. [ABSTRACT FROM AUTHOR]

In this work two kinds of smooth (compactons or cnoidal waves and solitons) and nonsmooth (peakons) solutions to the general Degasperis-Procesi (gDP) equation and its family (Degasperis-Procesi (DP) equation, modified DP equation, Camassa-Holm (CH) equation, modified CH equation, Benjamin-Bona-Mahony (BBM) equation, etc.) are reported in detail using different techniques. The single and periodic peakons are investigated by studying the stability analysis of the gDP equation. The novel compacton solutions to the equations under consideration are derived in the form of Weierstrass elliptic function. Also, the periodicity of these solutions is obtained. The cnoidal wave solutions are obtained in the form of Jacobi elliptic functions. Moreover, both soliton and trigonometric solutions are covered as a special case for the cnoidal wave solutions. Finally, a new form for the peakon solution is derived in details. As an application to this study, the fluid basic equations of a collisionless unmagnetized non-Maxwellian plasma is reduced to the equation under consideration for studying several nonlinear structures in the plasma model. [ABSTRACT FROM AUTHOR]